The dynamical sine-Gordon model in the full subcritical regime

TL;DR

This paper proves local well-posedness of the two-dimensional dynamical sine-Gordon equation across the full subcritical regime, overcoming previous limitations by leveraging charge cancellations for stochastic estimates.

Contribution

It introduces a novel approach using charge cancellations to handle non-polynomial nonlinearities and non-Gaussian noises, extending the well-posedness results to the entire subcritical regime.

Findings

Established local well-posedness for the full subcritical regime.

Developed stochastic estimates leveraging charge cancellations.

Extended previous partial results to the entire subcritical regime.

Abstract

We prove that the dynamical sine-Gordon equation on the two dimensional torus introduced in [HS16] is locally well-posed for the entire subcritical regime. At first glance this equation is far out of the scope of the local existence theory available in the framework of regularity structures [Hai14, BHZ16, CH16, BCCH17] since it involves a non-polynomial nonlinearity and the solution is expected to be a distribution (without any additional small parameter as in [FG17, HX18]). In [HS16] this was overcome by a change of variable, but the new equation that arises has a multiplicative dependence on highly non-Gaussian noises which makes stochastic estimates highly non-trivial - as a result [HS16] was only able to treat part of the subcritical regime. Moreover, the cumulants of these noises fall out of the scope of the later work [CH16]. In this work we systematically leverage "charge" cancellations specific to this model and obtain stochastic estimates that allow us to cover the entire subcritical regime.